IMPOWER function

The IMPOWER function in Excel raises a complex number to a specified power.

Syntax

IMPOWER(inumber, power)

Parameters

• inumber: The complex number that you want to raise to a power. The complex number can be entered as a string in the form a + bi or a + bj, where a is the real part and b is the imaginary part.
• power: The exponent to which you want to raise the complex number. It can be a positive or negative integer or a real number.

How It Works

To raise a complex number z = a + bi to a power n, Excel uses the following formula:

$$(a + bi)^n$$

This is calculated using polar coordinates and applying De Moivre’s theorem for complex numbers:

$$r^n \cdot (\cos(n\theta) + i \cdot \sin(n\theta))$$

Where:

• r is the modulus (absolute value) of the complex number, $r = \sqrt{a^2 + b^2}$
• θ is the argument (angle) of the complex number, $\theta = \tan^{-1}\left(\frac{b}{a}\right)$

Examples

1. Raising a Complex Number to a Power: To raise the complex number 2 + 3i to the power of 2:

   =IMPOWER("2+3i", 2)
   

   The result will be -5 + 12i.

2. Raising a Complex Number to a Negative Power: To raise the complex number 1 + 2i to the power of -2:

   =IMPOWER("1+2i", -2)
   

   The result will be 0.033613445378151 + -0.067226890756302i.

3. Raising a Complex Number to a Fractional Power: To raise the complex number 1 + 1i to the power of 0.5 (square root):

   =IMPOWER("1+1i", 0.5)
   

   The result will be approximately 1.09868411346781 + 0.45508986053271i.

4. Using a Cell Reference: If cell A1 contains the complex number "3+4i", you can raise it to the power of 3 using:

   =IMPOWER(A1, 3)
   

   The result will be approximately -117 + 44i.

Important Notes

• The IMPOWER function works only with valid complex numbers. If the input is not a valid complex number, Excel will return a #VALUE! error.
• The power parameter can be any real number, but complex numbers raised to non-integer powers may result in results that involve complex roots, depending on the exponent.
• Complex numbers in Excel can be represented using either i or j for the imaginary part.

Summary

The IMPOWER function in Excel raises a complex number to a specified power. This function is useful in advanced mathematical, engineering, and scientific calculations involving complex numbers, where raising complex numbers to various powers is required.